On the Carleman Ultradifferentiability of Weak Solutions of an Abstract Evolution Equation

Markin, Marat V.

doi:10.1007/978-3-7643-9921-4_26

Cited by 2 publications

(3 citation statements)

References 13 publications

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“…Here, we briefly outline certain facts essential for the subsequent discourse (for more details, see, e.g., [1][2][3]5]).…”

Section:  2 Preliminariesmentioning

confidence: 99%

“…[7,Ch. II,Definition 6.3], see also [5,Preliminaries]). • Such a notion of weak solution, which need not be differentiable in the strong sense, generalizes that of classical one, strongly differentiable on I and satisfying the equation in the traditional plug-in sense, the classical solutions being precisely the weak ones strongly differentiable on I.…”

Section: Weak Solutionsmentioning

confidence: 99%

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On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Markin

2020

Open Mathematics

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Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1 \beta \ge 1 , in particular analytic or entire, on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded. The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

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“…Here, we briefly outline certain facts essential for the subsequent discourse (for more details, see, e.g., [1][2][3]5]).…”

Section:  2 Preliminariesmentioning

confidence: 99%

Section: Weak Solutionsmentioning

confidence: 99%

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Markin

2020

Open Mathematics

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“…Ch. II, Definition 6.3], see also[24, Preliminaries]).The subspaces all possible initial values for the orbits under A, e tA t 0 { } ≥ , and e tA t { } ∈ are dense in X, ( ∥⋅∥) since they contain the subspace…”

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confidence: 99%

On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Markin

2020

Demonstratio Mathematica

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Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection { e t A } t ≥ 0 {\{{e}^{tA}\}}_{t\ge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C 0 {C}_{0} -semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group { e t A } t ∈ ℝ {\{{e}^{tA}\}}_{t\in {\mathbb{R}}} of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L 2 ( ℝ ) {L}_{2}({\mathbb{R}}) , the anti-self-adjoint differentiation operator A ≔ d d x A:= \tfrac{\text{d}}{\text{d}x} with the domain D ( A ) ≔ W 2 1 ( ℝ ) D(A):= {W}_{2}^{1}({\mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

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On the Carleman Ultradifferentiability of Weak Solutions of an Abstract Evolution Equation

Cited by 2 publications

References 13 publications

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

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